Full determination of single ferroelectric nanocrystal orientation by Pockels electro-Optic microscopy

Full determination of single ferroelectric nanocrystal orientation by Pockels electro-optic microscopy DUC THIEN TRINH,1,2 LUDOVIC MAYER,3 BASSAM HAJJ,1,4 JOSEPH LAUTRU,1 JOSEPH ZYSS,1 AND VASYL SHYNKAR1,* 1Laboratoire de photonique quantique et moléculaire, D’Alembert Institute, École Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94230 Cachan, France 2Faculty of Physics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam 3Laboratoire de Physique de la Matière Condensée, Ecole Polytechnique-CNRS, UMR 7643, 91128 Palaiseau Cedex, France 4Laboratoire Physico-Chimie, UMR 168, Institut Curie, 75005 Paris, France *Corresponding author: vasyl.shynkar@gmail.com Received 23 December 2014; revised 6 March 2015; accepted 11 March 2015; posted 12 March 2015 (Doc. ID 231105); published 9 April 2015 We present a nanoscale electro-optic imaging method allowing access to the phase response, which is not ame- nable to classical second-harmonic generation microscopy. This approach is used to infer the vectorial orientation of single domain ferroelectric nanocrystals, based on polarization-resolved Pockels microscopy. The electro-optic phase response of KTP nanoparticles yields the full orientation in the laboratory frame of randomly dispersed single nanoparticles, together with their electric polarization dipole. The complete vector determination of the dipole orientation is a prerequisite to important applications including ferroelectric nanodomain orientation, membrane potential imaging, and rotational dynamics of single biomolecules. © 2015 Optical Society of America OCIS codes: (180.3170) Interference microscopy; (160.2100) Electro-optical materials; (190.4400) Nonlinear optics, materials; (290.3030) Index measurements. 1. INTRODUCTION The past decades have been witnessing the development of a broad range of optical probing techniques down to the nano- scale. These advanced methods have allowed the investigation of various nano-objects, such as those made of organic aromatic molecules [1], as well as inorganic semiconductor quantum dots [2], resulting from various chemical synthesis [3] and nanofabrication [4] approaches. The latter have found many applications due to the stable and bright signal produced, and more importantly to well-established fabrication tech- niques for nanoparticles. The development of high-power fem- tosecond lasers and nonlinear second-harmonic generation microscopes (SHGMs) [5–7] has triggered further interest in the investigation of single nanocrystals, such as semiconductor quantum dots [8–11] and ferroelectric nonlinear optical par- ticles [12–16], with potential for bioimaging [12,17–20]. The latter domain opens the way to applications such as bright and photostable nanoscale sources, in view of the high contrast they provide with respect to their centrosymmetric background and the absence of photobleaching in nonresonant excitation conditions even under long-duration observations [21]. Among their advantages are their flexibility in the choice of excitation wavelength, a coherent signal for scanningless 3D imaging [12,22], a narrow signal bandwidth allowing for im- proved noise rejection, ultrafast response times, and biocom- patibility [23,24]. In order to fully exploit the information from such nanosources, it is necessary to determine their crys- talline quality and three-dimensional orientation. Although well established and widely used, SHGM suffers from impor- tant drawbacks, such as the need for a high-power ultra-short pulsed laser source so as to lead to the generation of a meas- urable second-harmonic signal, a relatively expensive near- infrared optics tool-box, as well as inherent difficulties in laser alignment and maintenance. The femtosecond Ti:Sa laser com- monly used in nanoscale characterization setups is expensive and bulky, difficult to align in the infrared, and the deposited energy can be destructive for objects of interest and, moreover, phototoxic in the case of biological samples. Searching for a different nonlinear second-order optical ef- fect could alleviate some of these drawbacks and make for an interesting alternative to SHGM. The linear electro-optical Pockels effect stands out as such an appealing alternative, also based on a quadratic nonlinearity [25] but with specific advan- tages of its own. As a result of the Pockels effect, the linear 3412 Vol. 54, No. 11 / April 10 2015 / Applied Optics Research Article 1559-128X/15/113412-10$15/0$15.00 © 2015 Optical Society of America variation of the refractive index of an optical medium upon an externally applied electric field can supply useful information on the crystalline structure at the nanoscale, as well as crystal orientation parameters. The Pockels effect has been widely used in optical devices to modulate a carrier beam [26]. It has recently found new applications, such as near-field optical microscopy for visualization of ferroelectric nanodomain dy- namics [27], measurements of electro-optic constants of semi- conductor crystals using a Mach–Zehnder interferometer setup [28], Pockels linear electro-optical microscopy (PLEOM) to- ward structural investigation of objects endowed with nonlinear optical properties, and electric field imaging with diffraction- limited resolution [29–32]. PLEOM has many advantages compared to SHGM. Electro-optical microscopy needs only a 1.5 mW CW He–Ne laser, basic optical elements in the vis- ible spectral range, a Mach–Zehnder interferometer configura- tion, simple Si photodiodes, and a lock-in device with standard electronics [29–32]. Moreover, PLEOM is an interferometric technique with higher sensitivity then SHGM, allowing mea- surement of the phase of the signal in addition to its amplitude, which compares advantageously to SHGM, which is limited to the determination of the amplitude of the harmonic power. It should also be mentioned that with PLEOM, in contrast to SHGM, the risk of phototoxicity is negligible due to the low power from the laser source. As a consequence, PLEOM can be safely applied to explore biological samples, for example, to- ward measurements of the cell plasma–membrane potential. We report here on the characterization of the nonlinear op- tical properties of single KTP nanocrystals via PLEOM. Using our knowledge of the externally applied electric field and of the electro-optic tensor of KTP crystals, we are able to retrieve the orientation of the crystal principal dielectric axis from the electro-optic scattering response of individual nanocrystals for different polarization states of the incident laser beam. The ad- vantage of this approach with respect to earlier methodology [29] and to SHGM [11,33–35] lies in its ability to lift the sign ambiguity on the crystal polarization along the z axis. Earlier methods relied on the measurement of the signal amplitude and could not reveal the signs of the electro-optical tensor coeffi- cients of the KTP crystal, leading to ambiguity in the measured crystal orientation with respect to the z axis. Additional information brought by the phase measurements from PLEOM provides the sign of the χ
2 coefficients as well as the full determination of the Euler angles of the nanocrystals. Such knowledge is of crucial importance when using nano- crystals toward electric field probing or bioimaging, using polarization-resolved microscopy to monitor the rotational dynamics of the investigated objects. 2. EXPERIMENT A. PLEOM Setup The PLEOM that was conceived and built in our laboratory [Fig. 1(a)] is based on a modified Mach–Zehnder interferom- eter, with a scanning microscope inserted in its probing arm in order to image the phase of the signal from the sample under investigation. The present microscope configuration is adapted to phase measurements of thin objects, in contrast to classical Mach–Zehnder interferometer for which phase shift through bulk materials can be measured. Our interferometric micro- scope was sensitive to small phase variations related to thermal fluctuations, as well as to mechanical vibrations. To filter out such spurious contributions and improve the signal-to-noise ratio (SNR), homodyne and synchronous detection techniques were applied to the detected signal. High-quality interference measurements were achieved by use of a 1.5 mW He–Ne sta- bilized laser with good spatial and temporal coherence. The laser beam was split into two: a high-intensity reference beam containing 90% of the original beam intensity, and a low- intensity signal beam with the remaining 10%. The beam was split by a polarizing beam splitting cube (PBS1). We also used a high-quality Glan–Thompson polarizer (Edmund Optics, USA, extinction rate 1,000,000:1) at the output of PBS1 to remove depolarized components from the reference and signal beams. In the signal arm of the interferometer, a commercial KD*P Pockels cell (Leysop, England) has been inserted, after the Glan–Thompson polarizer toward PLEOM calibration. During measurements, we applied an external electric field alternately to the Pockels cell and the sample. The measured values of the phase shifts introduced by the Pockels cell were used for calibration. A half-wave plate (Optique Fichou, Fresnes, France) was installed just after the Pockels cell to rotate the incident signal beam polarization and explore the sample polarization response. The polarized signal beam was focused Fig. 1. (a) General scheme of the PLEOM setup used to detect the EO signal. He–Ne laser, light source; PBS, polarization beam splitter; M, silver mirrors; PC, Pockels cell; O, objective (40×, 0.6 NA); λ∕2, half-wave plates; λ∕4, quarter-wave plate; P, photodiode. A lock-in amplifier used as a reference the signal from the generator. (b) Electron microscopy image of the KTP nanoparticle. Research Article Vol. 54, No. 11 / April 10 2015 / Applied Optics 3413 by a plan-fluor 40× (0.6 NA) Nikon objective with an extra- large working distance of 2.7–3.7 mm onto a diffraction- limited spot, subsequently used to scan the sample. The sample itself was set on a piezo stage (Piezojena, Jena, Germany) and moved by 200 nm steps under the focus spot with an integra- tion time of 100 ms. The signal beam with the phase shift in- troduced by the sample was then captured by a second “twin” objective, which shares its focus with that of the first objective. A phase shift is imprinted onto the signal beam by means of the Pockels effect. We applied an external sinusoidal electric field of variable amplitude from 0 to 150 V and constant fre- quency from 20 to 40 kHz, produced by a function generator (Tektronix, ES, France), both to the studied sample and to the KD*P Pockels cell. A second half-wave plate (Optique Fichou, Fresnes, France) was used at the output of the second objective to retrieve the orientation of the original beam polarization after passing through the microscope. This was achieved by keeping the orientation of the main axis of the second half-wave plate at the same position as the first one, during the polarization scan. The two beams were mixed inside the third polarization beam splitter (PBS3) and their corresponding projections were detected by Hamamatsu photodiodes. The amplified difference of the high-frequency signals mea- sured using the photodiodes was sent to a lock-in amplifier (EG&G Princeton Applied Research, Oak Ridge, Tennessee, USA). The lock-in amplifier used, as a reference, the signal from the generator, identical to that used for external electric field modulation, so as to achieve synchronous phase detection below the shot noise limit. The sensitivity of the interferometric detection of the electro-optic (EO) phase shift was maximized when the static optical path difference between the two beams lead to a phase shift of π∕2; a phase control loop, therefore, was implemented in our PLEOM. The phase control loop was built from mirrors mounted on a piezo-electric transducer stage (PI, Germany) driven by a low-frequency signal from the photode- tectors [Fig. 1(a)]. The lock-in amplifier was used to measure the amplitude and phase of the detected signal. The operation of electro-mechano-optical elements in the setup was synchron- ized and driven by LabWindows software, which was also ap- plied to create an image of the investigated sample from the lock-in acquired signals and the measured sample positions. B. Sample Preparation The samples used in these experiments were made of KTP nanoparticles dispersed on a glass coverslip with a system of planar gold electrodes evaporated on the glass surface and pat- terned by soft lithography techniques. The distance between the planar gold electrodes and the electrodes’ thickness were, respectively, 20 μm and 50 nm. A quasi-static electric field dis- tribution was created between the two planar electrodes by ap- plication of a sinusoidal voltage from the function generator, as shown in Fig. 1. A COMSOL simulation shows that the quasi- static electric field in the free space between the electrodes and near the surface of the glass is spatially uniform. We deposited on the substrate KTP nanocrystals synthesized and prepared as described by Mayer et al. [20]. Colloidal solutions containing KTP nanocrystals with a typical size of 150 nm, as measured by dynamic light scattering (DLS), were thus obtained. To obtain a sample for single nano-KTP measurements, the glass coverslip with the planar electrodes was initially dipped for 10 min in a 1% polydiallylmethyl-ammonium chloride (PDDAC) polymer solution (Sigma Aldrich) and then abundantly washed and dried. A few drops of nano-KTP solution were thus deposited onto the substrate and then washed and dried, leading to a sample embedding nanoparticles at the right concentration and distance toward single nanoparticle measurements. 3. THEORETICAL MODEL The light scattering properties of a nanoparticle depend on its size, and on the indices of refraction of both the nanoparticle and its environment. The refractive index of non-centrosym- metric crystalline nanoparticles further varies according to the Pockels electro-optic index modulation. Thus, in the case of nanoparticles, the scattered light can play the role of a Pockels effect indicator, whereas in bulk materials the directly transmitted light bears the information on the Pockels phase. Thus scattered light interferes with non-scattered light and the reference beam, leading to the following intensity [29]: I  jEωref  EωS  EωΩsc j2  I ref  I S  I sc  2Re
Eωref  EωS EωΩsc ; (1) where I ref , I S , and I sc are, respectively, the intensities of the reference Eωref , non-scattered E ω S , and scattered E ωΩ sc signal beams, which are fed into the control loop for PLEOM stability and for maintaining a constant π∕2 phase shift between the reference and the signal arms of the interferometer. The third term describes the interference, and contains the information about the Pockels phase. Indeed, in the case of nanoparticles excited by a focused laser beam in the presence of an external quasi-static electric field, the electric polarization of the nano- particle resulting from the electro-optical effect is given by the following expression: P
2Pockelsi
ω Ω ∝ ϵ0 X j X k rijk
ω ΩEωj EΩk ; (2) where
i; j; k are the indices related to the laboratory frame Oxyz, Eωj is the j component of the incident electric field, EΩk is the k component of the quasi-static low-frequency electric field applied through the electrodes, with modulation frequency Ω ≪ ω. rijk is the third-rank electro-optical Pockels tensor, which can be conventionally reduced to a two-dimensional array rij. KTP belongs to the mm2 point group, allowing us to sim- plify the rij matrix to the following form [25]: rij  0 BBBBB@ 0 0 r13 0 0 r23 0 0 r33 0 r42 0 r51 0 0 0 0 0 1 CCCCCA; (3) with r13  9.5 pm∕V, r23  15.7 pm∕V, r33  36.3 pm∕V, r42  9.3 pm∕V, and r51  7.3 pm∕V at 632.8 nm [36,37]. The electro-optical component of the nanoparticle polariza- tions, given by Eq. (2), oscillates at optical frequencies and leads to the following far-field scattered electromagnetic wave as follows: 3414 Vol. 54, No. 11 / April 10 2015 / Applied Optics Research Article Esc ∝ ∂2P ∂t2  −
ωΩ2P: (4) This last equation embodies all the needed information per- taining to the Pockels-induced phase change in KTP nanopar- ticles, together with the 3D orientation of the nanocrystals, as can be defined by the Euler angles
ψ ; θ; σ. The Euler angles define the orientation of the crystallographic axes
X ; Y ; Z  at- tached to the KTP nanoparticle with respect to the laboratory
x; y; z frame. For the mm2 point group, the crystallographic axes coincide with the principal dielectric axes, which simplify the analysis. The laboratory coordinate system Oxyz is defined so that the x axis coincides with the direction of the external quasi-static electric field and the z axis is perpendicular to the surface of the glass substrate. The external electric field EΩ  EΩ0 cos
Ωt  φE is oscillating at a low frequency close to 20 kHz with an initial phase φE . The polarization of the incident beam makes angle α with the x axis, leading to the following expression: Eω  Eω0
cos αux  sin αuy expi
ωt  φ0: (5) By using Eqs. (4) and (5) together with the transformation matrix corresponding to the Euler angles connecting the two coordinate systems, the expression for the interference term can be condensed into I signal ∝ −
A cos2 α B sin α cos α C sin2 α × EΩ0 cos
Ωt  φE ; (6) where A, B, and C are coefficients given by the trigonometric expression (see Appendix A) of the three Euler angles, also de- pending on components of the electro-optic tensor. Application of Eq. (6) to the description of the two configu- rations shown in Figs. 2(a) and 2(b) leads to the two following expressions for cases (a) and (b), respectively: I signal ∼
r33 cos2 α r13 sin2 α cos
Ωt  φE − π; (7) I signal ∼
r33 cos2 α r13 sin2 α cos
Ωt  φE: (8) From these expressions we observe that two oppositely ori- ented KTP nanoparticles feature electro-optical responses that are identical in amplitude and polarization [Fig. 2(c)], but opposite in their phases, thus evidencing that the amplitude response alone does not provide sufficient information to fully determine the orientation of such nanoparticles [29]. This limitation is particularly obvious for SHG microscopy. In what follows, we consequently use the polarization and phase responses of the measured signal to determine the full orienta- tion of individual nanocrystals. 4. RESULTS AND DISCUSSION We have investigated the nonlinear optical properties of KTP nanoparticles via quantitative PLEOM imaging. The ampli- tude and phase of the interference signal were measured and imaged at the scale of individual nanoparticles as already intro- duced. The images reflect the spatial distribution of the Pockels phase retardation imprinted onto the scattered light by the in- vestigated KTP nanocrystals. The Pockels phase retardation carried by the scattered light results from the application on the nanocrystals of a sinusoidal electric field of 5.104 V∕cm amplitude and 20 kHz frequency. The electric field, applied through gold electrodes, was modulated by an external voltage function generator. The detected lock-in amplitude, which ac- counts for the phase retardation of the scattered beam, was shown to be proportional to the amplitude of the externally applied electric field, thus ensuring that the response originated from the Pockels effect. We consequently raster scanned the sample by 20 nm steps to